Integrated Wishart bridge processes and generalised Hartman-Watson law

Jason Leung ¹¹1School of Mathematics and Statistics, e-mail: jason.leung@unimelb.edu.au ²²2Supported by the Albert Shimmins fund The University of Melbourne

(June, 2019)

Abstract

This article is concerned with the joint law of an integrated Wishart bridge process and the trace of an integrated inverse Wishart bridge process over the interval $\left[0,t\right]$ . Its Laplace transform is obtained by studying the Wishart bridge processes and the absolute continuity property of Wishart laws.

1 Introduction

Suppose $X$ is a solution to the stochastic differential equation on the cone $\tilde{\mathcal{S}}_{n}^{+}$ of $n\times n$ symmetric positive semi-definite matrices

\displaystyle dX_{t}=\sqrt{X}_{t}dW_{t}a+a^{\top}dW^{\top}_{t}\sqrt{X}_{t}+\left(bX_{t}+X_{t}b+\alpha a^{\top}a\right)dt,\quad t\geq 0,

(1.1)

where $X_{0}=x\in\tilde{\mathcal{S}}^{+}_{n}$ , $W$ is an $n\times n$ matrix-valued Brownian motion, $a$ in the space $GL(n)$ of $n\times n$ invertible matrices, $b$ in the cone $\tilde{\mathcal{S}}^{-}$ of negative semi-definite matrices such that $ab=ba$ and $\alpha\in\{1,2,\dots,n-1\}\cup(n-1,\infty)$ .

The process $X$ satisfying (1.1), first introduced in Bru (1991), is called a Wishart process of dimension $n$ , index $\alpha$ and parameters $a,b$ with initial value $x$ and is denoted $WIS(n,\alpha,a,b,x)$ . It was shown in Cuchiero et al. (2011) that the stochastic differential equation (1.1) has a unique weak solution for $\alpha\geq n-1$ as well as for $\alpha\in\{1,2,\dots,n-1\}$ with the additional condition of $\text{rank}(x)\leq\alpha$ . For $\alpha\geq n+1$ , Mayerhofer et al. (2011) showed that the solution to (1.1) exists as a strong solution and is unique for $t\geq 0$ . Moroever, it was shown in Mayerhofer et al. (2011) that if the initial value $x$ belongs to the space $\mathcal{S}_{n}^{+}$ of $n\times n$ positive definite matrices, the solution to (1.1) also belongs to $\mathcal{S}_{n}^{+}$ .

Given $\alpha\geq n+1$ and $x\in\tilde{\mathcal{S}}_{n}^{+}$ , then for $t\geq 0$ , the determinant of $X$ satisfies the stochastic differential equation

\displaystyle\begin{split}\ln\left(\det(X_{t})\right)&=\int_{0}^{t}\left(\alpha-n-1\right)Tr\left(a^{\top}aX^{-1}_{s}\right)+2Tr\left(b\right)ds\\ &\qquad+\int_{0}^{t}2Tr\left(\sqrt{X^{-1}_{s}}dW_{s}a\right),\qquad 0\leq t<\tau,\end{split}

(1.2)

where $\tau=\inf\left\{t\geq 0:\det{(X_{t})}=0\right\}$ . It was shown in Mayerhofer et al. (2011) Theorem 3.4 that for $\alpha\geq n+1$ and $x\in\mathcal{S}_{n}^{+}$ , $\tau=\infty$ almost surely.

Given $t\geq 0$ , the Laplace transform of $X_{t}$ can be computed directly from solving the matrix Riccati ordinary differential equation (see Ahdida and Alfonsi (2013) Proposition 4) and is given by

\displaystyle\mathbb{E}e^{-Tr\left(uX_{t}\right)}=\frac{\exp\left\{Tr\left[u\left(I-2\sigma_{t}u\right)^{-1}e^{bt}xe^{bt}\right]\right\}}{\det\left(I-2\sigma_{t}u\right)^{\alpha/2}},\quad u\in\tilde{\mathcal{S}}^{+}_{n},

where $\sigma_{t}=\int_{0}^{t}e^{bs}a^{\top}ae^{b^{\top}s}ds$ . Therefore, by comparing the above expression to the Laplace transform of the non-central Wishart random variable computed in Letac and Massam (2008), we deduce that $X_{t}$ follows the non-central Wishart distribution with $\alpha$ degrees of freedom, covariance matrix $\sigma_{t}$ and non-centrality matrix $e^{bt}xe^{bt}\sigma_{t}^{-1}$ , denoted $\mathcal{W}_{n}(\alpha,\sigma_{t},e^{bt}xe^{bt}\sigma_{t}^{-1})$ .

We denote the space of $n\times n$ matrix-valued continuous function defined on $[0,t]$ by $\mathcal{C}\left([0,\infty),\mathbb{R}^{n\times n}\right)$ , the law of a Wishart process $X$ on $\mathcal{C}\left([0,\infty),\mathbb{R}^{n\times n}\right)$ and its respective semi-group by ${}^{n}Q^{\alpha,a,b}_{x}$ , or simply $Q^{\alpha,a,b}_{x}$ when there is no ambiguities about the dimension $n$ . Moreover, we assume $\Omega=\mathcal{C}\left([0,\infty),\mathbb{R}^{n\times n}\right)$ , the set of $\mathbb{R}^{n\times n}$ -valued continuous functions defined on $[0,\infty)$ , and denote $X$ the coordinate process $X_{t}(\omega)=\omega_{t}$ .

For $\alpha\geq n+1$ , the Wishart law $Q^{\alpha,a,b}_{x}$ is absolutely continuous with respect to the parameters $\alpha$ and $b$ , their respective Cameron-Martin-Girsanov formulae are given as follows:

Lemma 1.1 (Absolute continuity of Wishart laws).

Let $\alpha\geq n+1$ , $t\geq 0$ and $Q^{\alpha,a,b}_{x}$ be the law of $WIS(n,\alpha,a,b,x)$ on $\mathcal{C}([0,\infty),\mathbb{R}^{n\times n})$ .

(i)

For $u\in\tilde{\mathcal{S}}_{n}^{-}$ such that $ua=au$ ,

\displaystyle\begin{split}dQ^{\alpha,a,b+u}_{x}=&\exp\left\{Tr\left[\frac{1}{2}(a^{\top}a)^{-1}u\left(X_{t}-X_{0}\right)\right.\right.\\ &\left.\left.-\frac{1}{2}\alpha ut-\int_{0}^{t}\left(a^{\top}a\right)^{-1}\left(u^{2}+bu\right)X_{s}ds\right]\right\}dQ^{\alpha,a,b}_{x}.\end{split}

(1.3)

(ii)

For $x\in\mathcal{S}^{+}_{n}$ and $\nu\in[\left(n+1-\alpha\right)/2,\infty)$ ,

\displaystyle\begin{split}dQ_{x}^{\alpha+2\nu,a,b}=&\left(\frac{\det X_{t}}{\det x}\right)^{\nu/2}\exp\bigg{\{}-Tr\bigg{[}\nu bt\\ &+\left(\alpha-n-1+\nu\right)\frac{\nu}{2}\int_{0}^{t}\left(a^{\top}a\right)X^{-1}_{s}ds\bigg{]}\bigg{\}}dQ^{\alpha,a,b}_{x}.\end{split}

(1.4)

Main result

This article is concerned with the joint conditional Laplace transform of the pair for

\displaystyle\left(\int_{0}^{t}X_{s}ds,\int_{0}^{t}Tr\left(a^{-1}aX^{-1}_{s}\right)ds\right),

(1.5)

for a $WIS(n,\alpha,a,b,x)$ process $X$ and $\alpha\geq n+1$ , given $X_{t}$ for a fixed $t\geq 0$ .

Let us first state the main result of this article,

Theorem 1.1.

Let $X$ be a $WIS(n,\alpha,a,b,x)$ process and $\alpha\geq n+1$ , then

\displaystyle\begin{split}&\mathbb{E}\left(\left.\exp\left\{-Tr\left[u^{2}\int_{0}^{t}X_{s}ds\right]-\frac{\lambda^{2}}{2}Tr\left[\int_{0}^{t}\left(a^{\top}a\right)X^{-1}_{s}ds\right]\right\}\right|X_{t}=y\right)\\ =&\frac{q^{\alpha+2\nu_{\lambda},a,b+\delta_{u}}_{t}(x,y)}{q^{\alpha,a,b}_{t}(x,y)}\left(\frac{\det y}{\det x}\right)^{-\nu_{\lambda}/2}\\ &\quad\exp\left\{Tr\left[\nu_{\lambda}bt+\left(\frac{1}{2}\left(a^{\top}a\right)^{-3/2}\left(u^{2}+bu\right)^{1/2}\left(y-x\right)-\alpha t\right)\right]\right\},\end{split}

(1.6)

where

	$\displaystyle u$	$\displaystyle\in\mathcal{D},\quad\lambda\in\mathbb{R},$
	$\displaystyle\delta_{u}$	$\displaystyle=\frac{1}{2}\left(-b+\sqrt{b^{2}-4a^{\top}au^{2}}\right),$
	$\displaystyle\mathcal{D}$	$\displaystyle=\left\{u\in\mathcal{S}_{n}:\delta_{u}+b\in\tilde{\mathcal{S}}_{n}^{+},au=ua\right\},$
	$\displaystyle\nu_{\lambda}$	$\displaystyle=\sqrt{\lambda^{2}+(\alpha-n-1)^{2}}-\alpha+n+1,$

and $q^{\alpha,a,b}_{t}(x,y)$ denotes the density of a $WIS(n,\alpha,a,b,x)$ semi-group.

Formula (1.6) is an extension of that given in Proposition 2.4 of Donati-Martin et al. (2004), where $a$ is assumed to be the identity matrix $\mathbf{id}$ and $b$ is $0$ . The proof for Theorem 1.1 relies on, as in that of Donati-Martin et al. (2004), the absolute continuity of Wishart law with respect to the dimension parameter $\alpha$ and the drift parameter $b$ as well as the law of a Wishart bridge process over $\left[0,t\right]$ , which will be defined in the next section.

2 Wishart bridge processes

A bridge of a Wishart process can be thought of as a Wishart process with its two end points “pinned down” over a fixed time interval. We define the law of a Wishart bridge process as a regular conditional probability measure, analogous to that of a squared Bessel bridge process as defined in Revuz and Yor (1999) Chapter XI.

We denote the space of $n\times n$ matrix-valued continuous function defined on $A\subseteq[0,\infty)$ by $\mathcal{C}\left(A,\mathbb{R}^{n\times n}\right)$ , the law of $X$ on $\mathcal{C}\left([0,\infty),\mathbb{R}^{n\times n}\right)$ and its respective semi-group by ${}^{n}Q^{\alpha,a,b}_{x}$ , or simply $Q^{\alpha,a,b}_{x}$ when there is no ambiguities about the dimension $n$ . Throughout this article, we assume $\Omega=\mathcal{C}\left([0,\infty),\mathbb{R}^{n\times n}\right)$ and denote $X$ the coordinate process $X_{t}(\omega)=\omega_{t}$ .

For every $t\geq 0$ , let us consider the space $\mathbb{W}_{t}=\mathcal{C}([0,t],\mathbb{R}^{n\times n})$ endowed with the topology generated by the uniform metric $\rho$ and the Borel $\sigma$ -algebra $\mathcal{B}(\mathbb{W}_{t})$ generated by this topology. Therefore the metric space $\left(\mathbb{W}_{t},\rho\right)$ is complete and separable (see Billingsley (1968)). Consequently, there exists a unique regular conditional distribution of ${}^{n}Q^{\alpha,a,b}_{x}({}\cdot|X_{t})$ , namely a family of probability measures ${}^{n}Q^{\alpha,a,b}_{x,y,t}$ on $\mathbb{W}_{t}$ such that for every $B\in\mathcal{B}$ ,

{}^{n}Q^{\alpha,a,b}_{x}(B)=\int{{}^{n}Q^{\alpha,a,b}_{x,y,t}}(B)\mu_{t}(dy),

where $\mu_{t}$ is the density of $X_{t}$ under ${}^{n}Q^{\alpha,a,b}_{x}$ .

Therefore we can define a Wishart bridge process by specifying its law as follow:

Definition 2.1.

A continuous process of which law is ${}^{n}Q^{\alpha,a,b}_{x,y,t}$ is called an $n$ -dimensional Wishart Bridge process (with parameters $\alpha,a,b$ ) from $x$ to $y$ over $[0,t]$ and is denoted by $WIS^{n,\alpha,a,b}_{t}(x,y)$ .

As for the law of a Wishart process, we simply write $Q^{\alpha,a,b}_{x,y,t}$ for the law of a Wishart bridge process when there is no ambiguities about the dimension. Loosely speaking, the law of a Wishart bridge can be understood in a sense that for every $B\in\mathcal{B}(\mathbb{W}_{t})$ ,

\displaystyle Q^{\alpha,a,b}_{x,y,t}(B)=Q^{\alpha,a,b}_{x}(B|X_{t}=y),

where $X$ is the coordinate process.

From the definition of a regular conditional probability (see, for example Ikeda and Watanabe (1989)), we observe that for every $B\in\mathcal{B}(\mathbb{W}_{t})$ , the map $y\mapsto Q^{\alpha,a,b}_{x,y,t}(B)$ is measurable and for every measurable function $f$ on $\mathbb{W}_{t}\times\mathbb{R}^{n\times n}$ ,

\displaystyle\int f(\omega,\omega_{t})Q^{\alpha,a,b}_{x}\left(d\omega\right)=\int\int f(\omega,y)Q^{\alpha,a,b}_{x,y,t}\left(d\omega\right)\mu_{t}\left(dy\right).

(2.1)

Throughout this article, we follow the notation in Revuz and Yor (1999) Chapter III, denoting a semi-group $P_{t}$ acting on an element $f$ in $\mathcal{C}_{0}\left(\mathbb{R}^{n\times n},\mathbb{R}\right)$ by $P_{t}f$ , that is

\displaystyle P_{t}f=\int f(y)P_{t}(x,dy),

where $\mathcal{C}_{0}\left(\mathbb{R}^{n\times n},\mathbb{R}\right)$ denotes the set of real-valued continuous functions on $\mathbb{R}^{n\times n}$ vanishing at infinity. And the function $p_{t}(x,y)$ such that

\displaystyle\int f(y)P_{t}(x,dy)=\int f(y)p_{t}(x,y)dy,

for every Borel measurable function $f$ is called the density of the semi-group $P_{t}$ .

We also make use of the square bracket $P_{t}\left[f\right]$ instead of $P_{t}\left(f\right)$ to avoid confusion with probability measures.

2.1 Integrated Wishart bridge processes

Suppose $X$ is a Wishart process with law $Q^{\alpha,a,b}_{x}$ , we call the process $Y$ defined by

\displaystyle Y_{t}=\int_{0}^{t}X_{s}ds,\quad t\geq 0,

an integrated Wishart process. An explicit formula for the conditional Laplace transform of $Y_{t}$ given $X_{t}$ at a fixed $t\geq 0$ was derived in Donati-Martin et al. (2004) for $\alpha\geq n+1$ , $a=\mathbf{id}$ and $b=0$ using the absolute continuity property of Wishart laws. Similarly, the aforementioned formula can be extended to a more general class of Wishart processes by using the absolute continuity property of Wishart laws.

Theorem 2.1.

Let $\alpha\geq n+1$ , $a\in GL(n)$ and $b\in\tilde{\mathcal{S}}_{n}^{-}$ be commutative. Then for $t\geq 0$ ,

	$\displaystyle Q_{x,y}^{b}\left[\exp\left\{-Tr\left[\left(u^{2}+bu\right)\int_{0}^{t}\left(a^{\top}a\right)^{-1}X_{s}ds\right]\right\}\right]$
	$\displaystyle=\frac{q_{t}^{b+u}(x,y)}{q_{t}^{b}(x,y)}\exp\left\{Tr\left[-\frac{1}{2}u\left(\left(a^{\top}a\right)^{-1}\left(y-x\right)-\alpha t\right)\right]\right\},\quad u\in\mathcal{D},$		(2.2)

where

\displaystyle\mathcal{D}=\left\{u\in\mathcal{S}_{n}:u+b\in\tilde{\mathcal{S}}_{n}^{-},au=ua\right\},

and $X$ is the coordinate process, $Q^{b}_{x,y}$ and $q_{t}^{b}$ denotes the $WIS^{n,\alpha,a,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,a,b,x)$ semigroup respectively.

Proof.

For every measurable Borel measurable function $f$ , it follows from (2.1) and the Cameron-Martin-Girsanov formula (1.3) that

		$\displaystyle\int Q_{x,y}^{b}\left[\exp\left\{-\int_{0}^{t}\left(a^{\top}a\right)^{-1}\left(u^{2}+bu\right)X_{s}ds\right\}\right]f(y)q^{b}_{t}(x,y)dy$
	$\displaystyle=$	$\displaystyle Q_{x}^{b}\left[\exp\left\{-\int_{0}^{t}\left(a^{\top}a\right)^{-1}\left(u^{2}+bu\right)X_{s}ds\right\}f\left(X_{t}\right)\right]$
	$\displaystyle=$	$\displaystyle Q^{b+u}_{x}\left[\exp\left\{Tr\left[-\frac{1}{2}u\left(\left(a^{\top}a\right)^{-1}\left(X_{t}-x\right)-\alpha t\right)\right]\right\}f(X_{t})\right]$
	$\displaystyle=$	$\displaystyle\int Q^{b+u}_{x,y}\left[\exp\left\{Tr\left[-\frac{1}{2}u\left(\left(a^{\top}a\right)^{-1}\left(X_{t}-x\right)-\alpha t\right)\right]\right\}\right]f(y)q^{b+u}_{t}(x,y)dy$
	$\displaystyle=$	$\displaystyle\int\exp\left\{Tr\left[-\frac{1}{2}u\left(\left(a^{\top}a\right)^{-1}\left(y-x\right)-\alpha t\right)\right]\right\}f(y)q^{b+u}_{t}(x,y)dy.$

Therefore, we have

		$\displaystyle Q_{x,y}^{b}\left[\exp\left\{-Tr\left(\int_{0}^{t}\left(u^{2}+bu\right)\left(a^{\top}a\right)^{-1}X_{s}\right)ds\right\}\right]q^{b}_{t}(x,y)$
	$\displaystyle=$	$\displaystyle\exp\left\{Tr\left[-\frac{1}{2}u\left(\left(a^{\top}a\right)^{-1}\left(y-x\right)-\alpha t\right)\right]\right\}q^{b+u}_{t}(x,y),$

almost surely. ∎

Replacing $u^{2}+bu$ in Theorem 2.1 with $u^{2}$ and solve

\displaystyle u^{2}=\left(a^{\top}a\right)^{-1}\left(\delta_{u}^{2}+b\delta_{u}\right),

for $\delta_{u}$ , we obtain the followings,

Corollary 2.1.

Let $\alpha\geq n+1$ , $a\in GL(n)$ and $b\in\tilde{\mathcal{S}}_{n}^{-}$ be commutative. Then for $t\geq 0$ ,

		$\displaystyle Q_{x,y}^{b}\left[\exp\left\{-Tr\left(u^{2}\int_{0}^{t}X_{s}ds\right)\right\}\right]$
		$\displaystyle=\frac{q_{t}^{b+\delta_{u}}(x,y)}{q_{t}^{b}(x,y)}\exp\left\{Tr\left[\frac{1}{2}(a^{\top}a)^{-3/2}\left(u^{2}+bu\right)^{1/2}u\left(y-x-\alpha t\right)\right]\right\},\quad u\in\mathcal{D},$		(2.3)

where

	$\displaystyle\delta_{u}$	$\displaystyle=\frac{1}{2}\left(-b+\sqrt{b^{2}-4a^{\top}au^{2}}\right),$
	$\displaystyle\mathcal{D}$	$\displaystyle=\left\{u\in\mathcal{S}_{n}:\delta_{u}+b\in\tilde{\mathcal{S}}^{-}_{n},au=ua\right\},$

and $X$ is the coordinate process, $Q^{b}_{x,y}$ and $q_{t}^{b}$ denotes the $WIS^{n,\alpha,a,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,a,b,x)$ semigroup respectively.

In the case of $WIS(n,\alpha,I_{n},0,x)$ , as considered in Donati-Martin et al. (2004), Corollary 2.1 allows us to find an explicit expression for the Laplace transform of an integrated Wishart bridge process. This extends formula (2.8) of Donati-Martin et al. (2004), where the Laplace transform of the trace of an integrated Wishart bridge process was considered. We summarise this result in the corollary below, which can also be considered as the matrix extension of formula (2.m) of Pitman and Yor (1982).

Corollary 2.2.

Let $\alpha\geq n+1$ . For every $u\in\tilde{\mathcal{S}}^{-}_{n}$ ,

\displaystyle Q_{x,y}\left[\exp\left\{-Tr\left(u^{2}\int_{0}^{t}X_{s}ds\right)\right\}\right]=\frac{q_{t}^{u}(x,y)}{q_{t}^{0}(x,y)}\exp\left(Tr\left[-\frac{u}{2}\left(\alpha t+x-y\right)\right]\right),

where $Q_{x,y}$ and $q^{b}_{t}(x,y)$ denote the $WIS^{n,\alpha,I,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,I,0,x)$ semigroup respectively.

2.2 Generalised Hartman-Watson law

The generalised Hartman-Watson law of a Wishart process $X$ for $a=\mathbf{id}$ and $b=0$ , namely the conditional distribution of

\displaystyle Tr\left(\int_{0}^{t}X^{-1}_{s}ds\right),

given $X_{t}$ , was studied in Donati-Martin et al. (2004) through its Laplace transform. By using the Wishart bridge processes and absolute continuity property of Wishart laws, the Laplace transform of the generalised Hartman-Watson law given in Donati-Martin et al. (2004) can also be obtained for $a\neq\mathbf{id}$ .

As in Theorem 2.1, by the definition of Wishart bridge processes and the Cameron-Martin-Girsanov formula (1.4), we have the followings,

Theorem 2.2.

Let $\alpha\geq n+1$ , $\nu\in[\left(n+1-\alpha\right)/2,\infty)$ and $t\geq 0$ , then

	$\displaystyle Q^{\alpha}_{x,y}$	$\displaystyle\left[\exp\left\{-\left(\alpha-n-1+\nu\right)\frac{\nu}{2}Tr\left[\left(a^{\top}a\right)\int_{0}^{t}X^{-1}_{s}ds\right]\right\}\right]$
		$\displaystyle=\frac{q_{t}^{\alpha+2\nu}(x,y)}{q_{t}^{\alpha}(x,y)}\left(\frac{\det y}{\det x}\right)^{-\nu/2}\exp\{\nu Tr(b)t\},$

where $Q^{\alpha}_{x,y}$ and $q^{\alpha}_{t}(x,y)$ denote the $WIS^{n,\alpha,a,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,a,b,x)$ semigroup respectively.

Proof.

As in the proof of Theorem 2.1 by applying (1.4) and (2.1). ∎

We can therefore compute the Laplace transform of the generalised Hartman-Watson law, which extends Proposition 2.4 of Donati-Martin et al. (2004) to a wider class of Wishart processes.

Corollary 2.3.

Let $\alpha\geq n+1$ and $t\geq 0$ , then for every $u\in\mathbb{R}$ ,

\displaystyle Q^{\alpha}_{x,y}\left[\exp\left\{-\frac{u^{2}}{2}Tr\left[\left(a^{\top}a\right)\int_{0}^{t}X^{-1}_{s}ds\right]\right\}\right]=\frac{q_{t}^{\alpha+2\nu_{u}}(x,y)}{q_{t}^{\alpha}(x,y)}\left(\frac{\det y}{\det x}\right)^{-\frac{\nu_{u}}{2}}\exp\{\nu_{u}Tr(b)t\},

where

\displaystyle\nu_{u}=\sqrt{u^{2}+(\alpha-n-1)^{2}}-\alpha+n+1,

$Q^{\alpha}_{x,y}$ and $q_{t}^{\alpha}$ denote the $WIS^{n,\alpha,a,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,a,b,x)$ semigroup respectively.

2.3 Proof of Theorem 1.1

Combining the arguments made in the proofs of Theorem 2.1 and Theorem 2.2, we have the following,

Theorem 2.3.

Let $a\in GL(n)$ , $b\in\tilde{\mathcal{S}}_{n}^{-}$ be commutative, $\alpha\geq n+1$ . Then for every $u\in\tilde{\mathcal{S}}_{n}$ such that $u+b\in\tilde{\mathcal{S}}_{n}^{-}$ , $ua=au$ and $\lambda\in\mathbb{R}$ ,

	$\displaystyle Q_{x,y,t}^{\alpha,a,b}\left[\exp\left\{-Tr\left[\left(u^{2}+bu\right)\int_{0}^{t}\left(a^{\top}a\right)^{-1}X_{s}ds\right]-\frac{\lambda^{2}}{2}Tr\left[\int_{0}^{t}\left(a^{\top}a\right)X^{-1}_{s}ds\right]\right\}\right]$
	$\displaystyle=\frac{q^{\alpha+2\nu_{\lambda},a,b+u}_{t}(x,y)}{q^{\alpha,a,b}_{t}(x,y)}\left(\frac{\det y}{\det x}\right)^{-\frac{\nu_{\lambda}}{2}}\exp\left\{Tr\left[\nu_{\lambda}bt-\frac{1}{2}u\left(\left(a^{\top}a\right)^{-1}\left(y-x\right)-\alpha t\right)\right]\right\},$

where

\displaystyle\nu_{\lambda}=\sqrt{\lambda^{2}+(\alpha-n-1)^{2}}-\alpha+n+1,

$Q^{b}_{x,y}$ and $q_{t}^{b}$ denotes the $WIS^{n,\alpha,a,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,a,b,x)$ semigroup respectively.

Therefore, Theorem 2.3 can be reformulated to give the joint Laplace transform of the pair

\displaystyle\left(\int_{0}^{t}X_{s}ds,\int_{0}^{t}Tr\left(a^{-1}aX^{-1}_{s}\right)ds\right),

under the Wishart bridge law.

Corollary 2.4.

Let $a\in GL(n)$ , $b\in\tilde{\mathcal{S}}_{n}^{-}$ be commutative, $\alpha\geq n+1$ . Then,

	$\displaystyle Q_{x,y,t}^{\alpha,a,b}$	$\displaystyle\left[\exp\left\{-Tr\left[u^{2}\int_{0}^{t}X_{s}ds\right]-\frac{\lambda^{2}}{2}Tr\left[\int_{0}^{t}\left(a^{\top}a\right)X^{-1}_{s}ds\right]\right\}\right]$
	$\displaystyle=$	$\displaystyle\frac{q^{\alpha+2\nu_{\lambda},a,b+\delta_{u}}_{t}(x,y)}{q^{\alpha,a,b}_{t}(x,y)}\left(\frac{\det y}{\det x}\right)^{-\nu_{\lambda}/2}$
		$\displaystyle\quad\qquad\exp\left\{Tr\left[\nu_{\lambda}bt+\left(\frac{1}{2}\left(a^{\top}a\right)^{-3/2}\left(u^{2}+bu\right)^{1/2}\left(y-x\right)-\alpha t\right)\right]\right\},$

where

	$\displaystyle u$	$\displaystyle\in\mathcal{D},\quad\lambda\in\mathbb{R},$
	$\displaystyle\delta_{u}$	$\displaystyle=\frac{1}{2}\left(-b+\sqrt{b^{2}-4a^{\top}au^{2}}\right),$
	$\displaystyle\mathcal{D}$	$\displaystyle=\left\{u\in\mathcal{S}_{n}:\delta_{u}+b\in\tilde{\mathcal{S}}_{n}^{-},au=ua\right\},$
	$\displaystyle\nu_{\lambda}$	$\displaystyle=\sqrt{\lambda^{2}+(\alpha-n-1)^{2}}-\alpha+n+1,$

$Q^{b}_{x,y}$ and $q_{t}^{b}$ denote the $WIS^{n,\alpha,a,b}_{t}(x,y)$ law and the density of a $WIS(n,\alpha,a,b,x)$ semi-group respectively.

Given a filtered probability space $\left(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right)_{t\geq 0},\mathbb{P}\right)$ and a Wishart process $X$ defined on it. Then Theorem 1.1 follows from Corollary 2.4 by identifying $\mathbb{E}\left({}\cdot{}|X_{t}=y\right)$ to $\mathbb{E}\left({}\cdot{}|\sigma(X_{t})\right)(\omega_{y})$ where $\omega_{y}\in\left\{\omega\in\Omega:X_{t}(\omega)=y\right\}$ .

References

(1)
Ahdida and Alfonsi (2013) Ahdida, A. and Alfonsi, A. (2013), ‘Exact and high-order discretization schemes for Wishart processes and their affine extensions’, The Annals of Applied Probability 23(3), 1025–1073.
Billingsley (1968) Billingsley, P. (1968), Convergence of Probability Measures, Wiley Series in probability and Mathematical Statistics: Tracts on probability and statistics, Wiley.
Bru (1991) Bru, M.-F. (1991), ‘Wishart processes’, Journal of Theoretical Probability 4(4), 725–751.
Cuchiero et al. (2011) Cuchiero, C., Filipović, D., Mayerhofer, E. and Teichmann, J. (2011), ‘Affine processes on positive semidefinite matrices’, The Annals of Applied Probability 21(2), 397–463.
Donati-Martin et al. (2004) Donati-Martin, C., Doumerc, Y., Matsumoto, H. and Yor, M. (2004), ‘Some properties of the Wishart processes and a matrix extension of the Hartman-Watson laws’, Publications of the Research Institute for Mathematical Sciences 40(4), 1385–1412.
Ikeda and Watanabe (1989) Ikeda, N. and Watanabe, S. (1989), Stochastic Differential Equations and Diffusion Processes, Kodansha scientific books, North-Holland.
Letac and Massam (2008) Letac, G. and Massam, H. (2008), ‘The noncentral Wishart as an exponential family, and its moments’, Journal of Multivariate Analysis 99(7), 1393 – 1417. Special Issue: Multivariate Distributions, Inference and Applications in Memory of Norman L. Johnson.
Mayerhofer et al. (2011) Mayerhofer, E., Pfaffel, O. and Stelzer, R. (2011), ‘On strong solutions for positive definite jump diffusions’, Stochastic Processes and their Applications 121(9), 2072 – 2086.
Pitman and Yor (1982) Pitman, J. and Yor, M. (1982), ‘A decomposition of Bessel bridges’, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 59(4), 425–457.
Revuz and Yor (1999) Revuz, D. and Yor, M. (1999), Continuous Martingales and Brownian Motion, Grundlehren der mathematischen Wissenchaften A series of comprehensive studies in mathematics, Springer.